The complement of an event is defined to be the set of all outcomes
contained in the sample space that are not contained in the event. It is
denoted by . Note that the complement of the sample space is defined to
be the empty set, , the set with no elements. Also,
and
for any event A. Therefore, if we set
,
, then is a countable collection
of mutually exclusive events. Hence, from axiom 3 we have,

Since
from Axiom 1, then this equation implies that
.

Note: mathematical equations are sentences with the same syntax as
English and can be read as such. The set operations, intersection,
union, and complement are often read as the English
equivalents, and, or, and not, respectively. Also,
the word or used in this context is assumed to mean the
inclusive or.

Now let
be a finite collection of mutually exclusive
events and set for . Then from Axiom 3, we have

That is, the probability of a finite union of mutually exclusive events equals
the sum of the probabilities.

Suppose we are interested in an experiment in which the sample space consists
of a finite collection of outcomes,
, and that
each outcome is equally likely with probability . Then the previous
result implies that

Therefore, we must have . Furthermore, since an event for such an
experiment may be written as the union of the individual outcomes contained
in the event, then

where represents the number of elements in the set A.
For experiments with equally likely outcomes, the probability of an event is
just the number of outcomes in the event divided by the total number of
outcomes.

Next note that A and are mutually exclusive and
. Therefore, from the previous result we have,

So, the probability of the complement of an event is one minus the probability
of the event. This result is useful for situations in which an event of
interest is very complicated and its probability is difficult to obtain
directly, but the complement of the event is simple with an easily obtainable
probability.

The axioms of probability tell us how to find the probability of the union
of mutually exclusive events, but not how to find the probability of the union
of arbitrary, not necessarily mutually exclusive, events. We can use the
results derived thus far to solve this problem. Suppose we are interested in
two events, A and B. We need to write the union of these
two events as the union of two mutually exclusive events. This can be done by
noting that
. Since A and
are mutually exclusive, then

Next note that
, which is a disjoint
union. Therefore,

and so,

Combining this with the previous result gives,

the probability of A union B equals the sum of the
probabilities minus the probability of the intersection.

In a similar way, we can show that probability is a monotone function.
Suppose that . Then we may express B as a disjoint
union,
and apply the additivity property of
probability,

since
. Hence, if , then .

Another extension that can be derived directly from the axioms is an
extremely useful result called the Law of Total Probability. A
partition of the sample space is defined to be a collection, finite
or countably infinite, of mutually exclusive events in the probability space
whose union is the sample space. Suppose that is partition and
A is an arbitrary event. Then
, and the
events, are mutually exclusive. The Law of Total Probability is
just the application of Axiom 3 to this expression,

This property allows us to breakdown a complicated event A into more
manageable pieces, .

Example. Suppose a standard card deck (13 denominations in 4 suits) is
well-shuffled and then the top card is discarded. What is the probability that
the card (the new top card) is an ace? Let enote the event that the
card is an ace. The partitioning events we will use are the events

Then,

The first term has numerator which is the number of ways the first card is an
ace and the second card is an ace, and has denominator which is the total
number of different outcomes for the first two cards. We can use permutations
to count the number of outcomes for both numerator and denominator. The
numerator is and the denominator is . Hence,

Similarly, the second term is

These give

Note that this probability is the same as the probability that the first card is
an ace.